Journal of Advanced Concrete Technology Vol. 2, No. 1, 77-87, February 2004 / Copyright 2004 Japan Concrete Institute 77

The Micro Truss Model: An Innovative Rational Design Approach for Reinforced Concrete Hamed M. Salem1

Received 16 January 2003, accepted 31 May 2003

Abstract The strut and tie models have been widely used as an effective tool for designing reinforced concrete structures. The concrete is considered to carry only compressive forces through, while the tension forces are carried by reinforcing steel. The strut and tie model is effective for designing disturbed regions, however, it is essential that the designer should have a minimum level of experience to assume optimum trusses. In this study, a generalization of the strut and tie model is introduced through the micro truss model, in which, small isotropic truss members are used and the macro strut and tie model are automatically obtained. Both material and geometrical nonlinearity are introduced. The proposed model can be used for both design and checking the nonlinear response of reinforced concrete structures. The model has been veri-fied through published experimental results. Rational steps of design have been incorporated and examples of design have been illustrated.

1. Introduction

1.1 Strut and tie models Strut and tie model is considered a rational and consis-tent basis for designing cracked reinforced concrete structures. It is mainly applied to the zones where the beam theory does not apply, such as geometrical discon-tinuities, loading points, deep beams and corbels. The approach is justified by the fact that reinforced concrete carries loads through a set of compressive stress fields, which are distributed and interconnected by tension ties. The ties may be reinforcing bars, prestressing tendons or concrete tensile stress fields. A sample of strut and tie model is shown in Fig. 1, which represents a continuous deep beam under point loading (MacGregor 1992)

Strut and tie models were firstly proposed by Ritter in 1899 as a simple truss model to visualize the internal forces in cracked beams. This model was the basis for Ritter (1899) and Morsch (1909) for the design of con-crete beams. Afterwards, it was refined by Kupfer (1964) and Leonhardt (1965). Marti (1985) created the scientific basis for a rational application in tracing the theory back to the theory of plasticity. Collins and Mitchell (1986) further considered the deformation of the truss model and derived a rational method for shear and torsion.

1.2 Lattice model In the lattice model, the continuum is discretized in a network of brittle beam or truss elements. The proce-dure was proposed in 1941 by Hrennikoff, who used large trusses to solve the problem of elasticity.

Herrmann (1991) applied the same model again for

modeling fracture. Herrmann used beam elements and fracture was simulated by removing beam elements as soon as specified failure strength was reached. The model proposed by Herrmann was linked to a finite element code by Vervuurt and Van Mier (1993). They used different arrangement of the lattice members, in which either a regular triangular lattice or a random lattice distribution was used. In both cases, the used lattice element was a beam element. The model was used on the micro level in order to simulate the fracture of concrete. Figure 2 shows a sample of modeling and analysis of lattice model by Vervuurt and Van Mier

1Asistant professor, Structural Engineering. Dept., Cairo University, Giza, Egypt. E-mail: hhadhoud@steelnetwork.com

Strut

Tie

Node

Anchorage Plate

Strut

Tie

Node

Anchorage Plate

Fig. 1 Sample of equilibrium strut-and-tie model (From MacGregor 1992).

Finite Elements ModelLattice Model

Aggregate

MatrixBond

Micro Lattice Model Tension Test Fig. 2 Lattice model (Vervuurt and Van Mier 1993).

78 H. M. Salem / Journal of Advanced Concrete Technology Vol. 2, No. 1, 77-87, 2004

(1993). The analysis represents axial tension test of plain concrete in which the aggregate, the surrounding matrix and the interfacial zone are reasonably simulated.

1.3 Modified lattice model Niwa et al. (1995) have developed another lattice model to explain the shear resisting mechanisms. That model is a macroscopic model in which the concrete is modeled into a flexural compression member, a flexural tension member, a diagonal compressive member, a diagonal tension member and an arch member. The reinforcement is modeled into horizontal and vertical members. The layout of the model is shown in Fig. 3. The main dif-ference between this model and the lattice model of Vervuurt (1993) is that this one is a macroscopic one while Vervuurts one is a microscopic model. The ratio of width of arch member to beam width t was deter-mined to minimize the total potential energy that is computed for a unit shear force acting on the concrete beam. The depth of the flexural compression member is made equal to the depth of the flexural compression zone at the flexural ultimate state. The depth of the flexural tension member is assumed to be twice the dis-tance between the centroid of the tensile reinforcement and the bottom fibers of the beam.

The height of the lattice model is assumed to be coin-cident with the effective depth of the beam. Thus, the diagonal members and the arch members are placed so as to connect the top surface of the beam and the cen-troid of the tensile reinforcing bars. The horizontal dis-tance of vertical members is assumed to equal half the effective depth. Therefore, the thickness of the truss member and the arch member are equal to (d/2) sin 45 and d sin , respectively where d is the depth of the

beam and is the inclination of the arch member. The model of Niwa (1995) predefines the tension

members, compression members and diagonal members. Niwa did not discuss whether his model is also applica-ble to deep beams, beams with openings, geometrical discontinuities or not.

1.4 Proposed model The proposed model adopts the conventional nonlinear analysis of trusses using the stiffness method. The nov-elty here is the application methodology itself. The proposed model is a microscopic model, similar to the lattice model of Vervuurt (1993). However, in the pro-posed model, the members are isotropically arranged, the stiffness of the members is calculated based on the dimensions of members, and fully nonlinear behavior is adopted for either concrete members or steel members. The objective of the present model is to simulate as well as to design reinforced concrete structures, which was not the goal of the lattice model of Vervuurt (1993). Therefore, in the present model the arrangement, the stiffness, the constitutive models and the objectives are dissimilar to lattice models ones.

It is also believed that the present model can be a generalization of Niwas model (Niwa 1995), which is a macroscopic model. Niwas model needs to predefine the compression members, the tension members and the dimensions of both depending on the beam theory. In a complicated structure, like deep beam with opening the author believes that Niwas model is not applicable. However, as will be illustrated later, the micro truss model is capable of analyzing general shape of struc-tures. The micro truss model is also more advantageous than Niwas model in the sense that the usage of small-size element enables the simulation of discrete cracks.

It is also believed that, a generalization of the strut and tie model can be introduced through the micro truss model. Micro truss model can automatically capture the macro struts and ties during analysis and could be help-ful to engineers to design complicated structures. 2. Formulation

2.1 The general form of the micro truss The micro truss model is a kind of generalization of the strut and tie model. The structure is divided into rela-tively large number of nodes that are connected by truss elements. The truss elements in fact represent the con-tinuum isotropically. Figure 4 shows the general form of the micro truss model. For each neighboring four nodes, there are two horizontal truss members, two ver-tical ones and two diagonal ones. The width of each member is assumed to equal the distance between the midway of the distance between the member and the two surrounding members (to its right and its left). The horizontal members carry the normal stresses in the horizontal direction while the vertical ones carry those

d

d/245

aV

ConcreteElement

SteelElement

ArchElement

b Flexural Compression Zone

Flexural TensionZone

Width ofTruss Member: b(1-t)/2Width of Arch

Member= bt

d

Fig. 3 Schematic diagram of the modified lattice model (Niwa et al. 1995).

d/2

b

=bt

b(1-t)/2

H. M. Salem / Journal of Advanced Concrete Technology Vol. 2, No. 1, 77-87, 2004 79

in the vertical direction. The diagonal members can transfer the shear through a mechanism of compressing one element and pulling the other. If the mesh is rotated 45 degrees, the role of the members is reversed. The horizontal and vertical members then carry shear load-ing while the diagonal members carry the normal stresses.

Therefore, the model is expected to simulate flexural cracks and diagonal cracks. In other words, it is ex-pected to simulate the flexural failure, the diagonal ten-sion failure, the shear-compression failure and the di-agonal splitting cracks. However, in the micro truss model, the aggregate interlock is not taken into consid-eration. Therefore, the proposed model cant simulate the sliding shear failure mode and may not be able to simulate the size effect for shear.

In Fig. 4 we can notice that the steel reinforcing bars are easily simulated. However, it should be kept in mind that the steel bars directions are limited to be horizontal, vertical and 45 degrees inclined. The author believes that, this limitation does not cause severe problems since practically most of the reinforcement bars are aligned as such. However, reinforcing bars can be aligned in any other directions by using anisotropic re-inforcement in horizontal and vertical directions respec-tively.

Full compatibility between steel and concrete at their interface is assumed. This assumption matches the real-ity for deformed bars, since the bar ribs interlock with the surrounding concrete and deform together (Okamura and Maekawa 1991). However, the slippage or the relative deformation between reinforcing bars and the far concrete, takes place as shown in Fig. 5.

2.2 Formulation of the stiffness matrix The global stiffness matrix of each truss member can be formulated directly by assuming unit displacement in the global directions as shown in Fig. 6. The elements of the stiffness matrix are represented as functions of the angle of inclination with the horizontal as follows,

[ ]

=

22

22

22

22

g

ss.css.cs.ccs.cc

ss.css.cs.ccs.cc

LEAS (1)

where c = cos, s = sin, E is the tangent stiffness of the stress-strain curve of the constituent material, A is the cross sectional area of the member and L is the length of the member. It should be mentioned that, the tangent stiffness has to be limited so that it is not zero or nega-tive in order to avoid divergence during the analysis. In fact a limiting minimum stiffness of 0.05 times the ini-tial stiffness is used here. Once the individual stiffness matrices [S]g for each member are determined, the over-all stiffness matrix [K] is assembled. 2.3 Material nonlinearity The element size in the micro truss model is chosen to be relatively small. Therefore, the constitutive laws adopted here should represent that micro level. In other words, the bare bar behavior and the plain concrete be-havior must be used. The concept of tension stiffening is meaningless here since it averages the behavior along relatively long gauge length containing several cracks.

Fig. 4 Schematic diagram of the micro-truss model.

Full compatibility with deformed bars at bar surfacemay be assumed

Flexural Cracks Propagate away from the bar

Deformed barsRelative elongation Slip

Fig. 5 Compatibility between steel and concrete at their interface.

Fig. 6 Formulation of the global stiffness matrix of the truss member.

1 =1

2 =1

3 =1 4 =1L

1=1

2

3=1 4

80 H. M. Salem / Journal of Advanced Concrete Technology Vol. 2, No. 1, 77-87, 2004

The use of such small-size elements in fact enables the simulation of discrete cracks, and the tension stiffening could be an outcome of the micro truss model. The be-havior of steel is a local point-wise behavior in which stress strain relationship of a bare bar is used. 2.3.1 Concrete Concrete in tension is simulated as plain concrete. After the concrete cracks the bridging tensile stress trans-ferred across the crack surface drops very fast. The re-sidual tensile strength is usually simulated as a post-cracking tension softening model. For this purpose, the post-cracking tension-stiffening model of Okamura (1991) is used with adjustment of the power coefficient C in order to apply the model effectively to the tension softening case. The model yields,

Ccr

tf

= (2)

where ft is the tensile strength of concrete, cr is the cracking strain and is the strain. The coefficient C is dependant on the fracture energy of plain concrete (Gf) as well as the size of the element. Considering that the plain concrete element is so small that it is expected to be crossed by only one crack as illustrated in Fig. 7, the residual stress-strain behavior after cracking is clearly dependant on the element length and the bridging stress transferred across the crack surface which, in its turn, depends on the fracture energy of concrete (Bazant and Oh 1983). The area under the stress-strain curve multi-plied by the element length represents the fracture en-ergy. Thus, the coefficient C is computed.

The fracture energy of plain concrete ranges from 0.1 to 0.15 N/mm (Uchida et al. 1991), and is kept constant regardless of element size.

Concrete in compression follows the elasto-plastic

and fracture model (Maekawa and Okamura 1983) as follows,

( )

( )peak

x35.0peakp

peak

c

e1x73.0

p

x

e1720x

f2Eo

eKo

)(EoKox25.1

=

=

==

=

(3)

where, Ko : Fracture parameter represents the damage of concrete, Eo : Initial Stiffness of concrete, p : Plastic strain corresponds the total strain , and peak : Peak strain for concrete under compression.

In fact, tension and compression models are not in-dependent with regard to their characteristic directions, but are mutually related in one way or another. How-ever, for simplicity the interaction among them is not considered in this study since the effect of hysteretic interaction is not so significant in monotonic loadings.

2.3.2 Reinforcing bar Reinforcing bar is simulated by Okamuras model for bare bars (1991). The stress is linear elastic up to yield-ing point and after a certain yielding plateau it starts strain hardening in an exponential form, as shown in Fig. 7.

( ) /

2/3

, 0,

{1 } (1.01 ) ,0.047(4000 / )

s y

y y sh

y u y sh

y

ksh

Ef

f e f fk f

=